thm:forbidden-tournaments-intro,thm:forbidden-tournaments-bad together imply thm:obstructions.
They also give lem:BST-twinwidth when applied to the class of tournaments with twin-width at most \(k\) :
this class can not contain any of \(\mathcal {F}_=,\mathcal {F}_\leqslant ,\mathcal {F}_\geqslant \) ,
hence its tournaments must still have bounded twin-width when paired with BST orders.
Finally, thm:obstructions,thm:forbidden-tournaments-bad directly imply that
if \(\mathcal {T}\)  is a hereditary class with unbounded twin-width,
then \(\mathcal {T}\)  satisfies none of the conditions of thm:main-thm.
The remaining implications of thm:main-thm—that is, when \(\mathcal {T}\)  has bounded twin-width, all other conditions hold—follow from known results on twin-width.
By [1]}, FO model checking has an \(\textrm {FPT}\)  algorithm
when a witness of bounded twin-width is given.
Combined with thm:approx-algo, we obtain an \(\textrm {FPT}\)  algorithm
for classes of tournaments with bounded twin-width.
By [1]}, a class of structures with bounded twin-width can not transduce all graphs.
Finally, by [3]}, a class of structures with bounded twin-width
contains at most \(c^n\)  structures on \(n\)  vertices up to isomorphism, for some constant \(c\)  depending on the class.
